Stable fluctuations for biased random walks on Galton–Watson trees

Consider a $\beta$-biased random walk with exponent $\alpha\in(1,2)$, where $\alpha$ is defined at the paper's equation labelled $\alpha_{\mathrm{tree}}$. Let $\Delta_n$ denote the hitting-time quantity used in the paper, and let $v$ be the speed. Stable-fluctuation conjecture. There should be no recentering and rescaling of the walk that yields scaling limits. Nevertheless, after recentering $\Delta_n$ by the inverse of the speed and rescaling by $n^{1/\alpha}$, the resulting sequence should be tight and should converge along exponential subsequences to an infinitely divisible distribution. The statement describes the expected non-Gaussian fluctuation regime for the tree walk; its precise formulation depends on the paper's definitions of $\Delta_n$, the speed, and $\alpha$.